I checked on the Quicklaunch design (https://vimeo.com/29822477) and the only solid moving mass in the gun is the projectile. So apparently you don't need to "bring [...] driver/piston to a halt extremely quickly" since there is none. They did propose to recover most of the gas by closing the muzzle just after the projectile exit, though. That might be complicated.
There is a piston. At 2:07 they show a close up of the first stage which they are referring to as the "pump tube", that larger diameter section, where the piston is driven up to speed, pumping the hydrogen into the second stage barrel at high pressure.
There is no piston. Where do you see one? Did you even watch the video? The whole launch sequence is shown there. There's a pressurized hot hydrogen tank that empties into the barrel. There's no "piston [...] driven up to speed, pumping the hydrogen". The hydrogen flows into the barrel on its own because it has high static pressure at the moment of firing. The presenter literally explains it in detail.
